Hindawi Publishing Corporation
International Journal of Stochastic Analysis
Volume 2010, Article ID 519684, 16 pages
doi:10.1155/2010/519684
Research Article
The Rothe’s Method to a Parabolic
Integrodifferential Equation with a Nonclassical
Boundary Conditions
Abdelfatah Bouziani and Rachid Mechri
Department of Mathematics, University of Larbi Ben M’Hidi, Oum El Bouagui 04000, Algeria
Correspondence should be addressed to Abdelfatah Bouziani, aefbouziani@yahoo.fr
Received 26 February 2009; Revised 17 August 2009; Accepted 10 December 2009
Academic Editor: Alexander M. Krasnosel’skii
Copyright q 2010 A. Bouziani and R. Mechri. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
This paper is devoted to prove, in a nonclassical function space, the weak solvability of parabolic
integrodifferential equations with a nonclassical boundary conditions. The investigation is made
by means of approximation by the Rothes method which is based on a semidiscretization of the
given problem with respect to the time variable.
1. Introduction
The purpose of this paper is to study the solvability of the following equation:
∂v
∂2 v
x, t − 2 x, t ∂t
∂x
t
at − sk s, vx, sds gx, t,
x, t ∈ 0, 1 × 0, T ,
1.1
0
with the initial condition
vx, 0 V0 x,
x ∈ 0, 1,
1.2
and the integral conditions
1
vx, tdx Et,
t ∈ 0, T ,
0
1
0
xvx, tdx Gt,
1.3
t ∈ 0, T ,
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where v is an unknown function, E, G, and V0 are given functions supposed to be sufficiently
regular, while k and a are suitably defined functions satisfying certain conditions to be
specified later and T is a positive constant.
Since 1930, various classical types of initial boundary value problems have been
investigated by many authors using Rothe time-discretization method; see, for instance, the
monographs by Rektorys 1 and Kačur 2 and references cited therein. The linear case of
t
our problem, that is, 0 at − sk s, vx, sds 0, appears, for instance, in the modelling
of the quasistatic flexure of a thermoelastic rod see 3 and has been studied, firstly, by
the second author with a more general second-order parabolic equation or a 2m-parabolic
equation in 3–5 by means of the energy-integrals method and, secondly, by the two authors
via the Rothe method 6–8. For other models, we refer the reader, for instance, to 9–12, and
references therein.
The paper is organized as follows. In Section 2, we transform problem 1.1–1.3 to
an equivalent one with homogeneous integral conditions, namely, problem 2.3. Then, we
specify notations and assumptions on data before stating the precise sense of the desired
solution. In Section 3, by the Rothe discretization in time method, we construct approximate
solutions to problem 2.3. Some a priori estimates for the approximations are derived in
Section 4, while Section 5 is devoted to establish the existence and uniqueness of the solution.
2. Preliminaries, Notation, and Main Result
It is convenient at the beginning to reduce problem 1.1–1.3 with inhomogeneous integral
conditions to an equivalent one with homogeneous conditions. For this, we introduce a new
unknown function u by setting
ux, t vx, t − Rx, t,
x, t ∈ 0, 1 × 0, T ,
2.1
where
Rx, t 62Gt − Etx − 23Gt − 2Et.
2.2
Then, the function u is seen to be the solution of the following problem:
∂2 u
∂u
x, t − 2 x, t ∂t
∂x
t
at − sks, ux, sds fx, t,
x, t ∈ 0, 1 × 0, T ,
0
ux, 0 U0 x,
1
ux, tdx 0,
x ∈ 0, 1,
t ∈ 0, T ,
0
1
0
xux, tdx 0,
t ∈ 0, T ,
2.3
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3
where
fx, t gx, t −
∂Rx, t
,
∂t
U0 x V0 x − Rx, 0,
2.4
ks, ux, s k s, ux, s − Rx, t.
Hence, instead of looking for the function v, we search for the function u. The solution of
problem 1.1–1.3 will be simply given by the formula vx, t ux, t Rx, t.
We introduce the function spaces, which we need in our investigation. Let L2 0, 1
and L2 0, T ; L2 0, 1 be the standard function spaces. We denote by C0 0, 1 the linear space
of continuous functions with compact support in 0, 1. Since such functions are Lebesgue
integrable, we can define on C0 0, 1 the bilinear form given by
u, v 1
Ix uIx v dx,
2.5
uζ, ·dζ.
2.6
0
where
Ix u x
0
The bilinear form 2.5 is considered as a scalar product on C0 0, 1 for which C0 0, 1 is not
complete.
Definition 2.1. We denote by B21 0, 1 a completion of C0 0, 1 for the scalar product 2.5,
which is denoted by ·, ·B21 0,1 , called the Bouziani space or the space of square integrable
primitive functions on 0, 1. By the norm of function u from B21 0, 1, we understand the
nonnegative number
uB21 0,1 u, uB21 0,1 Ix u,
2.7
where v denotes the norm of v in L2 0, 1.
For u ∈ L2 0, 1, we have the elementary inequality
1
uB21 0,1 ≤ √ u.
2
2.8
We denote by L2 0, T ; B21 0, 1 the space of functions which are square integrable in
the Bochner sense, with the scalar product
u, vL2 0,T ;B21 0,1 T
0
u·, t, v·, tB21 0,1 dt.
2.9
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Since the space B21 0, 1 is a Hilbert space, it can be shown that L2 0, T ; B21 0, 1 is a
Hilbert space as well. The set of all continuous abstract functions in 0, T equipped with the
norm
2.10
sup u·, τB21 0,1
0≤τ≤T
is denoted C0, T ; B21 0, 1. Let V be the set which we define as follows:
V v ∈ L2 0, 1;
1
vxdx 1
0
xvxdx 0 .
2.11
0
Since V is the null space of the continuous linear mapping l: L2 0, 1 → R2 , ϕ →
1
1
lϕ 0 ϕxdx, 0 xϕxdx, it is a closed linear subspace of L2 0, 1, consequently V is a
Hilbert space endowed with the inner product ·, ·. Strong or weak convergence is denoted
by → or , respectively. The letter C will stand for a generic positive constant which may be
different in the same discussion.
Lemma 2.2 Gronwall’s lemma. a1 Let xt ≥ 0, ht, yt be real integrable functions on the
interval a, b. If
yt ≤ ht t
xsysds,
∀t ∈ a, b,
2.12
a
then
yt ≤ ht t
t
hsxs exp
a
xτdτ ds,
∀t ∈ 0, T .
2.13
a
In particular, if xt ≡ C is a constant and ht is nondecreasing, then
yt ≤ htect−a ,
∀t ∈ 0, T .
2.14
a2 Let {ai }i be a sequence of real nonnegative numbers satisfying
i−1
ai ≤ A Bh ak ,
∀i 1, 2, . . . ,
2.15
k1
where A, B, and h are positive constants, such that Bh < 1. Then
ai ≤ A expBi − 1h,
takes place for all i 1, 2, . . . .
2.16
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5
In the sequel, we make the following assumptions.
H1 Functions f : 0, T → L2 0, 1 and a : 0, T → R are Lipschitz continuous, that
is,
ft − f t ≤ l1 t − t , ∀t ∈ 0, T ,
∃l2 ∈ R ; at − a t ≤ l2 t − t , ∀t ∈ 0, T .
∃l1 ∈ R ;
2.17
H2 The mapping k : 0, T × V → L2 0, 1 is Lipschitz continuous in both variables,
that is,
kt, u − k t , u ≤ l3 t − t u − u ,
∃l3 ∈ R ;
2.18
for all t, t ∈ I, u, u ∈ V , and satisfies
∃l4 , l5 ∈ R ;
kt, uB21 0,1 ≤ l4 uB21 0,1 l5 ,
2.19
for all t ∈ I and all u ∈ V , where l4 and l5 are positive constants.
H3 U0 ∈ H 2 0, 1 and
1
U0 xdx 1
0
xU0 xdx 0.
2.20
0
We will be concerned with a weak solution in the following sense.
Definition 2.3. A function u : I → L2 0, 1 is called a weak solution to problem 2.3 if the
following conditions are satisfied:
i u ∈ L∞ I, V ∩ CI, B21 0, 1,
ii u is strongly differentiable a.e. in I and du/dt ∈ L∞ I, B21 0, 1,
iii u0 U0 in V ,
iv the identity
du
ut, v
t, v
dt
B21 0,1
t
at − sks, usds, v
0
ft, v
B21 0,1
2.21
B21 0,1
holds for all v ∈ V and a.e. t ∈ 0, T .
To close this section, we announce the main result of the paper.
Theorem 2.4. Under assumptions H1 –H3 , problem 2.3 admits a unique weak solution u, in
the sense of Definition (2.3).
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3. Construction of an Approximate Solution
In order to solve problem 2.3 by the Rothe method, we proceed as follows. Let n be a
positive integer, we divide the time interval I 0, T into n subintervals Ijn : tnj−1 , tnj ,
j 1, . . . , n, where tnj : jhn and hn : T/n. Then, for each n ≥ 1, problem 2.3
may be approximated by the following recurrent sequence of time-discretized problems.
Successively, for j 1, . . . , n, we look for functions unj ∈ V such that
unj − unj−1
hn
−
d2 unj
dx2
hn
j−1 a tnj − tni k tni , uni fjn ,
3.1
i0
1
unj xdx 0,
3.2
xunj xdx 0,
3.3
0
1
0
starting from
un0 U0 ,
δun0 d2
U0 f0,
dx2
3.4
where unj x : ux, tnj , δunj : unj − unj−1 /hn , fjn x : fx, tnj . For this, multiplying for all
x ξ
j 1, . . . , n, 3.1 by I2x v : 0 0 vτdτdξ and integrating over 0, 1, we get
1
0
δunj xI2x v dx −
1 d 2 un
j
0
dx2
xI2x v dx hn
1 1
j−1 a tnj − tni k tni , uni I2x v dx fjn I2x v dx.
0 i0
0
3.5
Noting that, using a standard integration by parts, we have
I21 v
1
1 − ξvξdξ 0
1
vξdξ −
1
0
ξvξdξ 0,
∀v ∈ V.
3.6
0
Carrying out some integrations by parts and invoking 3.6, we obtain for each term in 3.5
1
0
δunj I2x v dx − δunj , v
B21 0,1
1 d 2 un
j
0
hn
dx2
,
xI2x v dx unj , v ,
3.7
1 j−1 j−1 a tnj − tni k tni , uni x I2x v dx −hn a tnj − tni k tni , uni , v B1 0,1 ,
0 i0
i0
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7
and for the last one
1
0
fjn xI2x vxdx − fjn , v
B21 0,1
3.8
.
By virtue of 3.7 and 3.8, 3.5 becomes
δunj , v
B21 0,1
j−1 unj , v hn a tnj − tni k tni , uni , v B1 0,1 fjn , v
B21 0,1
2
i0
,
3.9
or
unj , v
B21 0,1
h2n
hn unj , v
j−1 a tnj − tni k tni , uni , v B1 0,1 hn fjn , v
i0
B21 0,1
2
3.10
unj−1 , v 1
.
B2 0,1
Let η·, · : V × V → R and Lj · : V → R be two functions defined by
ηu, v u, vB21 0,1 hn u, v,
Lj v h2n
j−1 a tnj − tni k tni , uni , v B1 0,1 hn fjn , v
i0
B21 0,1
2
unj−1 , v
3.11
B21 0,1
.
It is easy to see that the bilinear form η·, · is continuous on V and V-elliptic, and
the form Lj · is continuous for each j 1, . . . , n. Then, Lax-Milgram lemma guarantees the
existence and uniqueness of unj , for all j 1, . . . , n.
4. A Priori Estimates
Lemma 4.1. There exists C > 0 such that, for all n ≥ 1 and all j 1, . . . , n, the solution uj of the
discretized problem 3.1–3.4 satisfies the estimates
n
uj ≤ C,
n
≤ C.
δuj 1
B2 0,1
4.1
4.2
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International Journal of Stochastic Analysis
Proof. Testing the difference 3.9j−1 -3.9j with v δunj ∈ V , taking into account assumptions
H1 –H3 and the Cauchy-Schwarz inequality, we obtain
n
δuj B21 0,1
unj − unj−1 B21 0,1
≤ δunj−1 j−2
C1 2 C1
C1 hn uni B1 0,1 hn hn unj−1 1
,
2
B2 0,1
3
3
3
i0
B21 0,1
4.3
where
C1 : 3 max{l2 ζ, T l2 ζ M1 ζ l1 },
M1 : max|at|,
t∈I
ζ : max{l4 , l5 }.
4.4
Multiplying the left-hand side of the last inequality with 1 − C1 /3hn < 1 and adding the
terme
2
C1 hn unj − unj−1 1
− δunj 1
4.5
< 0,
B2 0,1
B2 0,1
3
we get
1 − C1 hn δunj unj δunj−1 B21 0,1
≤ unj−1 B21 0,1
B21 0,1
B21 0,1
j−2
C1 h2n uni B1 0,1 C1 hn .
4.6
2
i0
Applying the last inequality recursively, it follows that
1 − C1 hn j δunj B21 0,1
unj B21 0,1
j−2
≤ un0 B1 0,1 δun0 B1 0,1 C1 T T C1 hn uni B1 0,1 ,
2
2
i0
4.7
2
or, by virtue of Lemma 2.2, there exists n0 ∈ N∗ such that
n
δuj B21 0,1
unj B21 0,1
≤ C2 ,
∀n ≥ n0 ,
4.8
where
C2 : expT C1 1 δun0 B1 0,1 un0 B1 0,1 T C1
2
× exp expT C1 1 T C1 ,
and so our proof is complete.
2
4.9
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We address now the question of convergence and existence.
5. Convergence and Existence
Now let us introduce the Rothe function un t : I → V obtained from the functions uj by
piecewise linear interpolation with respect to time
un t unj−1 δunj t − tnj−1 ,
5.1
in Ijn ,
u
n t, fn t, and kt,
n t defined as follows:
as well the step functions u
n t, u
u
n t ⎧
⎨un0 ,
⎩un ,
j
for t 0,
in Ijn : tnj−1 , tnj ,
fn t for t 0,
⎩un ,
j−1
in Ijn ,
⎧
⎨f0, for t 0,
⎩f n ,
j
kn t u
n t ⎧
⎨un0 ,
5.3
in Ijn ,
⎧
0,
⎪
⎪
⎨
5.2
for t 0,
j−1 ⎪
n
n
n
n
⎪
⎩hn a tj − ti k ti , ui ,
in Ijn tnj−1 , tnj .
5.4
i0
Corollary 5.1. There exist C > 0 such that the estimates
un t ≤ C, ∀t ∈ I,
un t ≤ C, n du
≤ C, for a.e. t ∈ I,
dt t
1
B2 0,1
un t − un tB21 0,1 ≤ Chn ,
un t − un tB21 0,1 ≤ Chn , kn t
≤ C, ∀t ∈ I,
5.5
5.6
∀t ∈ I,
5.7
5.8
hold for all n ∈ N∗ .
Proof. For the inequalities 5.5, 5.6, and 5.7 see 6, Corollary 4.2., whereas for the last
inequality, assumption H2 and estimate 4.1 guarantee the desired result.
Proposition 5.2. The sequence un n converges in the norm of the space CI, B21 0, 1 to some
function u ∈ CI, B21 0, 1 and the error estimate
un − uCI,B21 0,1 ≤ C hn
takes place for all n ≥ n0 .
5.9
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Proof. By virtue of 5.2, 5.3, and 5.4 the variational equation 3.9 may be rewritten in the
form
dun
t, v
dt
B21 0,1
un t, v kn t, v
B21 0,1
fn t, v
B21 0,1
,
5.10
for a.e. t ∈ 0, T . In view of 5.10, using 5.6 and 5.8 with the fact that
n f t
≤ M2 : max
ft
B1 0,1 < ∞,
B21 0,1
t∈I
2
5.11
we obtain
un t, v| ≤
|
kn t
B21 0,1
≤ CvB21 0,1 ,
n du
t
vB21 0,1
B21 0,1
dt
B21 0,1
fn t
5.12
a.e. t ∈ 0, T .
Now, for n, m being two positive integers, testing the difference 5.10n -5.10m with v un t − um t which is in V, with the help of the Cauchy-Schwarz inequality and taking into
account that
d
d
ut, ut
ut2B1 0,1 ,
2
2
1
dt
dt
B2 0,1
a.e. t ∈ 0, T ,
5.13
and, by virtue of 5.12 we obtain after some rearrangements
1 d n
un t − u
m t2
u t − um t2B1 0,1 2
2 dt
≤ Cum t − u
un t − un tB21 0,1
m tB21 0,1 C
kn t − km t
1
un t − um tB21 0,1
5.14
B2 0,1
fn t − fm t
B21 0,1
un t − um tB21 0,1 ,
a.e. t ∈ 0, T .
To derive the required result, we need to estimate the third and the last terms in the righthand side, for this, let t be arbitrary but fixed in 0, T , without loss of generality we can
suppose that there exist three positive integers p, q and β, such that
m
t ∈ tnp−1 , tnp ∩ tm
q−1 , tq ,
n βm, tnp tm
q .
5.15
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Hence, using 5.4 we can write
kn t − km t
B21 0,1
⎡
⎤
β j1 −1
p−1 n n n n m
m
m
⎣
⎦
a tp − tj k tj , uj − a tm
hm q − ti k ti , ui
j0
ijβ
.
B21 0,1
5.16
m
By virtue of assumption H1 and the fact that |atnp − tnj − atm
q − ti | ≤ Chn , there exist
εn ∈ 0, Chn such that
kn t − km t
≤ hm
p−1
j0
B21 0,1
⎡
βj1−1
⎣
Chn − εn k tnj , unj B21 0,1
ijβ
m m m n
n
t
−
k
ti , ui a tm
−
t
,
u
k
q
i
j
j
⎤
B21 0,1
⎦.
5.17
Therefore, recalling assumptions H1 , H2 and having in mind that εn ∈ 0, Chn , we
estimate
kn t − km t
B21 0,1
≤ hm
p−1
⎡
βj1−1
⎣
j0
Chn C hn unj − um
i B21 0,1
ijβ
⎤
⎦,
5.18
m
from where, we derive for all s ∈ tm
i , ti1 kn t − km t
B21 0,1
≤ hm
p−1
⎡
βj1−1
⎣
j0
ijβ
Chn C hn un s − un sB21 0,1
⎤
5.19
un s − um sB21 0,1 um s − u
m sB21 0,1 ⎦.
Taking the supremum with respect to s from 0 to t in the right-hand side, invoking the fact
m
n
n
that s ∈ tm
i , ti1 ⊂ tj−1 , tj and estimate 5.7, we obtain
kn t − km t
B21 0,1
≤ hm
q−1
i0
Chn Csup u s − u sB21 0,1 ,
n
m
5.20
0≤s≤t
so that
kn t − km t
B21 0,1
≤ Chn Csup un s − um sB21 0,1 .
0≤s≤t
5.21
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International Journal of Stochastic Analysis
m
Let t ∈ tnp−1 , tnp ∩ tm
q−1 , tq , from assumption H1 it follows that
n
f t − fm t
B21 0,1
f tnp − f tm
q ≤ l1 tnp − tm
q B21 0,1
5.22
≤ l1 hn .
Ignoring the second term in the left-hand side of 5.14 which is clearly positive and using
estimates 5.5, 5.7, 5.21, and 5.22 yield
d n
u t − um t2B1 0,1 ≤ Chn hm Csup un s − um s2B1 0,1 ,
2
2
dt
0≤s≤t
a.e. t ∈ 0, T .
5.23
Integrating this inequality with respect to time from 0 to t and invoking the fact that un 0 um 0 U0 , we get
u t − u
n
m
t2B1 0,1
2
≤ Chn hm C
t
sup un ξ − um ξ2B1 0,1 dξ,
5.24
sup un ξ − um ξ2B1 0,1 dξ.
5.25
2
0 0≤ξ≤t
whence
sup un s − um s2B1 0,1 ≤ Chn hm C
2
0≤s≤t
t
0 0≤ξ≤t
2
Accordingly, by Gronwall’s lemma we obtain
sup un s − um s2B1 0,1 ≤ Chn hm expCt,
2
0≤s≤t
∀t ∈ 0, T ,
5.26
consequently
sup un s − um sB21 0,1 ≤ C hn hm
5.27
0≤s≤t
takes place for all n, m ∈ N∗ . This implies that un tn is a Cauchy sequence in the Banach
space CI, B21 0, 1, and hence it converges in the norm of this latter to some function u ∈
CI, B21 0, 1. Besides, passing to the limit m → ∞ in 5.27, we obtain the desired error
estimate, which finishes the proof.
Now, we present some properties of the obtained solution.
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13
The limit-function u from Proposition 5.2, possesses the following properties:
i u ∈ CI, B21 0, 1 ∩ L∞ I, V ,
ii u is strongly differentiable a.e. in I and du/dt ∈ L∞ I, B21 0, 1,
iii u
n t → ut in B21 0, 1 for all t ∈ I,
n t ut in V for all t ∈ I,
iv un t, u
v dun /dtt du/dtt in L2 I, B21 0, 1.
Proof. On the basis of estimates 5.5 and 5.6, uniform convergence statement from
Proposition 5.2, and the continuous embedding V → B21 0, 1, the assertions of the present
theorem are a direct consequence of 2, Lemma 1.3.15.
Theorem 5.3. Under Assumptions H1 –H3 , 2.3 admits a unique weak solution, namely, the
limit function u from Proposition 5.2, in the sense of Definition 2.3.
Proof. We have to show that the limit function u satisfies all the conditions i, ii, iii,
and iv of Definition 2.3. Obviously, in light of the properties of the function u listed in
Theorem 5.3, the first two conditions of Definition 2.3 are already seen. On the other hand,
since un → u in CI, V as n → ∞ and, by construction, un 0 U0 , it follows that u0 U0 ,
so the initial condition is also fulfilled, that is, Definition 2.3iii takes place. It remains to see
that the integral identity 2.21 is obeyed by u. For this, integrating 5.10 over 0, t and using
the fact that un 0 U0 , we get
un t − U0 , vB21 0,1 t
t u
kτ,
n τ, v
un τ, vdτ 0
0
B21 0,1
dτ t fn τ, v
0
B21 0,1
dτ,
5.28
consequently, after some rearrangements
un t − U0 , vB21 0,1 t τ
0
t
un τ, vdτ
0
aτ − sks, usds, v
t u
kτ,
n τ −
τ
0
B21 0,1
0
dτ t
0
fn τ − fτ, v
aτ − sks, usds, v
0
2
5.29
0
t fτ, v B1 0,1 dτ
B21 0,1
dτ
B21 0,1
dτ.
Let sn : I → I and sn : I → I denote the functions
sn t ⎧
⎨0,
for t 0,
⎩tn ,
j−1
in Ijn ,
sn t ⎧
⎨0,
for t 0,
⎩tn ,
in Ijn .
j
5.30
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International Journal of Stochastic Analysis
To investigate the desired result, we prove some convergence statements. Using 5.2, 5.4,
and 5.30 we have for all t ∈ tnj−1 , tnj u
kt,
n t −
t
at − sks, usds
0
5.31
tn
tn j
j
sn s, u
a tnj − sn s k
at − sks, usds.
n s − at − sks, us ds 0
t
Taking into account 5.5, 5.9, and assumptions H1 , H2 it follows that
n
sn s, u
n s − at − sks, us
a tj − sn s k
B21 0,1
≤ C hn .
5.32
Thanks to 5.31 and 5.32 we obtain
t
n t − at − sks, usds
kt, u
0
≤ C hn .
5.33
B21 0,1
On the other hand, in view of the assumed Lipschitz continuity of f, we have
n
f τ − fτ
B21 0,1
sn τ − fτ
B1 0,1
≤ f
2
5.34
≤ l1 hn .
u
Now, the sequences {
un τ, v}, {fn τ, vB1 0,1 }, and {kτ,
n τ, vB21 0,1 } are uniformly
2
bounded with respect to both τ and n, so the Lebesgue theorem of majorized convergence is
applicable to 5.29. Thus, having in mind 5.7, 5.9, 5.33, and 5.34, we derive that
ut − U0 , vB21 0,1 t τ
0
t
uτ, vdτ
0
aτ − sks, usds, v
0
B21 0,1
dτ t
0
fτ, v B1 0,1 dτ
5.35
2
takes place for all v ∈ V and t ∈ 0, T . Finally, differentiating 5.35 with respect to t, we get
d
ut, v
ut, v
dt
B21 0,1
t
at − sks, usds, v
0
ft, v
B21 0,1
B21 0,1
5.36
,
a.e. t ∈ 0, T .
International Journal of Stochastic Analysis
15
The uniqueness may be argued in the usual manner. Indeed, exploiting an idea in 11,
consider u1 and u2 two different solutions of 2.3, and define w u1 − u2 then, we have
t
d
wt, v
wt, v at − sks, u1 s − ks, u2 sds, v
. 5.37
dt
0
B21 0,1
1
B2 0,1
Choosing v wt as a test function, with the aid of Cauchy-Schwarz inequality and
assumption H1 , we obtain
1 d
wt2B1 0,1 wt2 ≤ C
2
2 dt
t ks, u1 s − ks, u2 sB21 0,1 dswtB21 0,1 .
5.38
0
Let ξ ∈ 0, p such that
wξB21 0,1 max wsB21 0,1 ,
s∈0,p
5.39
integrating 5.38 over 0, p, 0 ≤ p ≤ T , using 5.39, and invoking assumption H2 , we get
p
0
1 d
2
2
wtB1 0,1 wt dt ≤ Cp2 wξ2B1 0,1 ,
2
2
2 dt
5.40
consequently, with the fact that w0 0
p
0
ξ
d
1 d
wt2B1 0,1 wt2 dt ≤ Cp2
wt2B1 0,1 dt.
2
2
2 dt
dt
0
5.41
Choosing p as a constant verifying the condition
∃α ∈ N,
T αp,
Cp2 ≤
1
,
2
5.42
we have, by virtue of 5.41
p
1 d
wt2B1 0,1 dt 2
2
dt
0
p
wt2 dt ≤
0
ξ
1 d
wt2B1 0,1 dt,
2
2
dt
0
5.43
taking into account that ξ ≤ p, we obtain
wt 0,
on 0, p .
5.44
Following the same lines as for 0, p, we deduce that
wt 0,
on ip, i 1p , i 1, 2, 3, . . . ,
therefore, we derive wt ≡ 0, on 0, T , then u1 ≡ u2 . This achives the proof.
5.45
16
International Journal of Stochastic Analysis
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